Abstract

Theoretical and experimental studies of the diffraction of a two-dimensional reflection grating are performed in this paper. Based on the theory of optical scattering, the light field in the Fraunhofer diffraction region is deduced, and the general expression of the diffraction field is given in the form of the wave vectors of the diffracted wave and the incident wave. Then the coordinate of the diffraction order is obtained. The calculation results show that the diffraction distortion of the grating appears when the grating is illuminated by the oblique incident light wave and the distortion is restricted on the diffraction of the grids varying along the direction perpendicular to the plane of incidence. The orbit equation satisfied by the distortion diffraction orders is presented. The experimental results verify adequately this diffraction distortion rule of the grating, and they agree very well with the theoretical results.

© 2009 Optical Society of America

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References

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  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
    [CrossRef]
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    [CrossRef]
  3. K. Busch,G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. Rev. Lett. 444, 101-202 (2007).
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    [CrossRef]
  5. S. Teng, Y. Tan, and C. Cheng, “Quasi-Talbot effect of the high-density grating in near field,” J. Opt. Soc. Am. A 25, 2945-2951 (2008).
    [CrossRef]
  6. J. E. Harvey and R. V. Shack, “Aberrations of diffracted wave fields,” Appl. Opt. 17, 3003-3009 (1978).
    [CrossRef] [PubMed]
  7. J. E. Harvey and C. L. Vernold, “Description of diffraction grating behavior in direction cosinusoidal space,” Appl. Opt. 37, 8158-8160 (1998).
    [CrossRef]
  8. J. E. Harvey, D. Bogunovic, and A. Krywonos, “Aberrations of diffracted wave fields: distortion,” Appl. Opt. 42, 1167-1174(2003).
    [CrossRef] [PubMed]
  9. G. Fortin, “Graphical representation of the diffraction grating equation,” Am. J. Phys. 76, 43-47 (2008).
    [CrossRef]
  10. J. A. Ogilvy, Theory of Wave Scattering from Rough Surfaces (Hilger, 1991).
  11. J. Wang, L. Fang, Z. Zhang, J. Du, and Y. Guo, “Implementation of coherent array beam combination using two dimensional sine phase grating,” Chin. J. Laser 35, 39-43(2008).
    [CrossRef]
  12. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Function with Formulas, Graphs, and Mathematical Tables (Government Printing Office, 1965).

2008 (5)

J. J. Zheng, C. H. Zhou, and B. Wang, “Phase interpretation for polarization-dependent near-field images of high-density gratings,” Opt. Commun. 281, 3254-3259 (2008).
[CrossRef]

G. Fortin, “Graphical representation of the diffraction grating equation,” Am. J. Phys. 76, 43-47 (2008).
[CrossRef]

J. Wang, L. Fang, Z. Zhang, J. Du, and Y. Guo, “Implementation of coherent array beam combination using two dimensional sine phase grating,” Chin. J. Laser 35, 39-43(2008).
[CrossRef]

D. Crouse, A. P. Hibbins, and M. J. Lockyear, “Tuning the polarization state of enhanced transmission in gratings,” Appl. Phys. Lett. 92, 191105 (2008).
[CrossRef]

S. Teng, Y. Tan, and C. Cheng, “Quasi-Talbot effect of the high-density grating in near field,” J. Opt. Soc. Am. A 25, 2945-2951 (2008).
[CrossRef]

2007 (1)

K. Busch,G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. Rev. Lett. 444, 101-202 (2007).

2003 (1)

1998 (2)

J. E. Harvey and C. L. Vernold, “Description of diffraction grating behavior in direction cosinusoidal space,” Appl. Opt. 37, 8158-8160 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

1978 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Function with Formulas, Graphs, and Mathematical Tables (Government Printing Office, 1965).

Bogunovic, D.

Busch, K.

K. Busch,G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. Rev. Lett. 444, 101-202 (2007).

Cheng, C.

Crouse, D.

D. Crouse, A. P. Hibbins, and M. J. Lockyear, “Tuning the polarization state of enhanced transmission in gratings,” Appl. Phys. Lett. 92, 191105 (2008).
[CrossRef]

Du, J.

J. Wang, L. Fang, Z. Zhang, J. Du, and Y. Guo, “Implementation of coherent array beam combination using two dimensional sine phase grating,” Chin. J. Laser 35, 39-43(2008).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

Fang, L.

J. Wang, L. Fang, Z. Zhang, J. Du, and Y. Guo, “Implementation of coherent array beam combination using two dimensional sine phase grating,” Chin. J. Laser 35, 39-43(2008).
[CrossRef]

Fortin, G.

G. Fortin, “Graphical representation of the diffraction grating equation,” Am. J. Phys. 76, 43-47 (2008).
[CrossRef]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

Guo, Y.

J. Wang, L. Fang, Z. Zhang, J. Du, and Y. Guo, “Implementation of coherent array beam combination using two dimensional sine phase grating,” Chin. J. Laser 35, 39-43(2008).
[CrossRef]

Harvey, J. E.

Hibbins, A. P.

D. Crouse, A. P. Hibbins, and M. J. Lockyear, “Tuning the polarization state of enhanced transmission in gratings,” Appl. Phys. Lett. 92, 191105 (2008).
[CrossRef]

Krywonos, A.

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

Linden, S.

K. Busch,G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. Rev. Lett. 444, 101-202 (2007).

Lockyear, M. J.

D. Crouse, A. P. Hibbins, and M. J. Lockyear, “Tuning the polarization state of enhanced transmission in gratings,” Appl. Phys. Lett. 92, 191105 (2008).
[CrossRef]

Mingaleev, S.

K. Busch,G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. Rev. Lett. 444, 101-202 (2007).

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Rough Surfaces (Hilger, 1991).

Shack, R. V.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Function with Formulas, Graphs, and Mathematical Tables (Government Printing Office, 1965).

Tan, Y.

Teng, S.

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

Tkeshelashvili, L.

K. Busch,G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. Rev. Lett. 444, 101-202 (2007).

Vernold, C. L.

von Freymann, G.

K. Busch,G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. Rev. Lett. 444, 101-202 (2007).

Wang, B.

J. J. Zheng, C. H. Zhou, and B. Wang, “Phase interpretation for polarization-dependent near-field images of high-density gratings,” Opt. Commun. 281, 3254-3259 (2008).
[CrossRef]

Wang, J.

J. Wang, L. Fang, Z. Zhang, J. Du, and Y. Guo, “Implementation of coherent array beam combination using two dimensional sine phase grating,” Chin. J. Laser 35, 39-43(2008).
[CrossRef]

Wegener, M.

K. Busch,G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. Rev. Lett. 444, 101-202 (2007).

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

Zhang, Z.

J. Wang, L. Fang, Z. Zhang, J. Du, and Y. Guo, “Implementation of coherent array beam combination using two dimensional sine phase grating,” Chin. J. Laser 35, 39-43(2008).
[CrossRef]

Zheng, J. J.

J. J. Zheng, C. H. Zhou, and B. Wang, “Phase interpretation for polarization-dependent near-field images of high-density gratings,” Opt. Commun. 281, 3254-3259 (2008).
[CrossRef]

Zhou, C. H.

J. J. Zheng, C. H. Zhou, and B. Wang, “Phase interpretation for polarization-dependent near-field images of high-density gratings,” Opt. Commun. 281, 3254-3259 (2008).
[CrossRef]

Am. J. Phys. (1)

G. Fortin, “Graphical representation of the diffraction grating equation,” Am. J. Phys. 76, 43-47 (2008).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

D. Crouse, A. P. Hibbins, and M. J. Lockyear, “Tuning the polarization state of enhanced transmission in gratings,” Appl. Phys. Lett. 92, 191105 (2008).
[CrossRef]

Chin. J. Laser (1)

J. Wang, L. Fang, Z. Zhang, J. Du, and Y. Guo, “Implementation of coherent array beam combination using two dimensional sine phase grating,” Chin. J. Laser 35, 39-43(2008).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

Opt. Commun. (1)

J. J. Zheng, C. H. Zhou, and B. Wang, “Phase interpretation for polarization-dependent near-field images of high-density gratings,” Opt. Commun. 281, 3254-3259 (2008).
[CrossRef]

Phys. Rep. Rev. Lett. (1)

K. Busch,G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. Rev. Lett. 444, 101-202 (2007).

Other (2)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Function with Formulas, Graphs, and Mathematical Tables (Government Printing Office, 1965).

J. A. Ogilvy, Theory of Wave Scattering from Rough Surfaces (Hilger, 1991).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the scattering theory for analyzing the grating diffraction.

Fig. 2
Fig. 2

Positions of the diffraction orders of a 300 line per mm grating with respect to the grids varying with y 0 .

Fig. 3
Fig. 3

Orbit curves for the diffraction orders with different incident angles.

Fig. 4
Fig. 4

Schematic diagram of the experimental measurement of the grating diffraction.

Fig. 5
Fig. 5

Experimental result of the diffraction pattern of the two-dimensional reflection grating.

Fig. 6
Fig. 6

Schematic diagram of the relationship of the coordinate and orientation angles of the observation point.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

x = z f sin θ 2 sin θ 3 cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos θ 3 ,
y = z f cos θ 2 sin θ 1 cos θ 1 sin θ 2 cos θ 3 cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos θ 3 ,
U ( k , k ) = F A ( a h x 0 + b h y 0 c ) exp { i [ k · r 0 + k h ( r 0 ) ] } d 2 r 0 ,
k | | = k 0 ( sin θ 2 cos θ 3 sin θ 1 ) x ^ 0 + k 0 sin θ 2 sin θ 3 y ^ 0 ,
k = k 0 ( cos θ 1 + cos θ 2 ) ,
U ( k , k | | ) = F c A exp { i [ k | | · r 0 + k h ( r 0 ) ] } d 2 r 0 .
h ( r 0 ) = h 0 [ sin ( 2 π x 0 d ) + sin ( 2 π y 0 d ) ] ,
U ( k , k | | ) = F c A exp [ i ( k x x 0 + k y y 0 ) ] exp [ i k h 0 ( sin 2 π x 0 d + sin 2 π y 0 d ) ] d x 0 d y 0 ,
U ( k , k | | ) = B m , n J n ( k h 0 ) J m ( k h 0 ) δ ( k x + 2 π n d ) δ ( k y + 2 π m d ) ,
sin θ 2 = [ sin 2 θ 1 + ( m λ ) 2 / d 2 ] 1 / 2 ,
sin θ 3 = m λ d [ sin 2 θ 1 + ( m λ ) 2 / d 2 ] 1 / 2 .
x = z f m λ / { d cos θ 1 [ cos 2 θ 1 ( m λ ) 2 / d 2 ] 1 / 2 + d sin 2 θ 1 } ,
y = z f sin θ 1 cos θ 1 sin θ 1 [ cos 2 θ 1 ( m λ ) 2 / d 2 ] 1 / 2 cos θ 1 [ cos 2 θ 1 ( m λ ) 2 / d 2 ] 1 / 2 + sin 2 θ 1 .
x 2 sin 2 θ 1 = y 2 cos 2 θ 1 y z f sin 2 θ 1 .
r sin θ 2 cos θ 3 = z f sin θ 1 y cos θ 1 .
r cos θ 2 = z f cos θ 1 + y sin θ 1 .
y = z f cos θ 2 sin θ 1 cos θ 1 sin θ 2 cos θ 3 cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos θ 3 ,
r = z f 1 cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos θ 3 .
x = z f sin θ 2 sin θ 3 cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos θ 3 .
[ cos 2 θ 1 ( m λ ) 2 / d 2 ] 1 / 2 = y sin 2 θ 1 cos θ 1 + z f sin θ 1 cos θ 1 z f sin θ 1 y cos θ 1 .
( m λ ) 2 / d 2 = cos 2 θ 1 ( y sin 2 θ 1 cos θ 1 + z f sin θ 1 cos θ 1 z f sin θ 1 y cos θ 1 ) 2 .
x 2 = z f 2 cos 2 θ 1 ( y sin 2 θ 1 cos θ 1 + z f sin θ 1 cos θ 1 z f sin θ 1 y cos θ 1 ) 2 ( cos θ 1 y sin 2 θ 1 cos θ 1 + z f sin θ 1 cos θ 1 z f sin θ 1 y cos θ 1 + sin 2 θ 1 ) 2 .
x 2 = y 2 cos 2 θ 1 y z f sin 2 θ 1 sin 2 θ 1 ,
x 2 sin 2 θ 1 = y 2 cos 2 θ 1 y z f sin 2 θ 1 .

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